Of abelian subcategories

He first modifies the notion of the defining sheaf of ideals of a closed subscheme to the notion of defining ideal of a topologizing subcategory as the endofunctor ℐ=ℐS∈End(A)\mathcal{I}=\mathcal{I}_S\in End(A) which is the subfunctor of identity IdAId_A assigning to any M∈AM\in A the intersection of kernels Ker(f)Ker(f) of all morphisms f:M→Nf: M\to N where N∈Ob(S)N\in Ob(S). One can show that if T⊂ST\subset S is an inclusion of topologizing subcategories, then ℐS⊂ℐT\mathcal{I}_{S}\subset \mathcal{I}_{T}. In particular, for Gabriel multiplication of topologizing subcategories we have ℐS∘S⊂ℐS\mathcal{I}_{S\circ S} \subset \mathcal{I}_S.

Then the conormal bundle is simply ΩS=ℐS/ℐS∘S\Omega_S = \mathcal{I}_S/\mathcal{I}_{S\circ S}, similarly to the sheaf case.